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1. The Model 57
central bank is zero inflation in America. In case B the targets of the European central bank are zero inflation and zero unemployment in Europe. And the targets of the American central bank are zero inflation and zero unemployment in America. In case C the European central bank has a single target, that is zero inflation in Europe. By contrast, the American central bank has two conflicting targets, that is zero inflation and zero unemployment in America. This chapter deals with case A, and the next chapters deal with cases B and C.
The target of the European central bank is zero inflation in Europe. The instrument of the European central bank is European money supply. By equation (3), the reaction function of the European central bank is:
2M1 = − 2B1 + M2 (5)
Suppose the American central bank lowers American money supply. Then, as a response, the European central bank lowers European money supply.
The target of the American central bank is zero inflation in America. The instrument of the American central bank is American money supply. By equation (4), the reaction function of the American central bank is:
2M2 = −2B2 + M1 (6)
Suppose the European central bank lowers European money supply. Then, as a response, the American central bank lowers American money supply.
The Nash equilibrium is determined by the reaction functions of the European central bank and the American central bank. The solution to this problem is as follows:
3M1 = −4B1 −2B2 (7) 3M2 = −4B2 −2B1 (8)
Equations (7) and (8) show the Nash equilibrium of European money supply and American money supply. As a result there is a unique Nash equilibrium. According to equations (7) and (8), an increase in B1 causes a decline in both
58 Monetary Interaction between Europe and America: Case A
European money supply and American money supply. A unit increase in B1 causes a decline in European money supply of 1.33 units and a decline in
American money supply of 0.67 units.
From equations (1), (7) and (8) follows the equilibrium rate of unemployment in Europe:
u1 = A1 + B1 (9)
From equations (2), (7) and (8) follows the equilibrium rate of unemployment in America:
u2 = A2 + B2 (10)
From equations (3), (7) and (8) follows the equilibrium rate of inflation in Europe:
π1 = 0 (11)
And from equations (4), (7) and (8) follows the equilibrium rate of inflation in America:
π2 = 0 (12)
As a result, given a shock, monetary interaction produces zero inflation in Europe and America.
2. Some Numerical Examples 59
2. Some Numerical Examples
For easy reference, the basic model is summarized here:
u1 = A1 −M1 +0.5M2 (1) u2 = A2 −M2 +0.5M1 (2) π1 = B1 + M1 −0.5M2 (3) π2 = B2 + M2 −0.5M1 (4)
And the Nash equilibrium can be described by two equations:
3M1 = −4B1 −2B2 (5) 3M2 = −4B2 −2B1 (6)
It proves useful to study six distinct cases: - a demand shock in Europe
- a supply shock in Europe - a mixed shock in Europe
- another mixed shock in Europe - a common demand shock
- a common supply shock.
1) A demand shock in Europe. In each of the regions, let initial unemployment be zero, and let initial inflation be zero as well. Step one refers to a decline in the demand for European goods. In terms of the model there is an
increase in A1 of 3 units and a decline in B1 of equally 3 units. Step two refers to the outside lag. Unemployment in Europe goes from zero to 3 percent.
Unemployment in America stays at zero percent. Inflation in Europe goes from zero to – 3 percent. And inflation in America stays at zero percent.
Step three refers to the policy response. According to the Nash equilibrium there is an increase in European money supply of 4 units and an increase in
60 Monetary Interaction between Europe and America: Case A
American money supply of 2 units. Step four refers to the outside lag. Unemployment in Europe goes from 3 to zero percent. Unemployment in America stays at zero percent. Inflation in Europe goes from – 3 to zero percent. And inflation in America stays at zero percent. Table 3.1 presents a synopsis.
Table 3.1
Monetary Interaction between Europe and America A Demand Shock in Europe
Europe America
Unemployment 0 Inflation 0
Shock in A1 3 Shock in B1 − 3 Unemployment 3
Inflation − 3 Change in Money Supply 4 Unemployment 0
Inflation 0
Unemployment 0 Inflation 0
Unemployment 0 Inflation 0 Change in Money Supply 2 Unemployment 0
Inflation 0
As a result, given a demand shock in Europe, monetary interaction produces zero inflation and zero unemployment in each of the regions. The loss functions of the European central bank and the American central bank are respectively:
L1 = π1 (7) L2 = π2 (8)
The initial loss of the European central bank is zero, as is the initial loss of the American central bank. The demand shock in Europe causes a loss to the European central bank of 9 units and a loss to the American central bank of zero
2. Some Numerical Examples 61
units. Then monetary interaction reduces the loss of the European central bank from 9 to zero units. And what is more, monetary interaction keeps the loss of the American central bank at zero units.
2) A supply shock in Europe. In each of the regions let initial unemployment be zero, and let initial inflation be zero as well. Step one refers to the supply
shock in Europe. In terms of the model there is an increase in B1 of 3 units and an increase in A1 of equally 3 units. Step two refers to the outside lag. Inflation in Europe goes from zero to 3 percent. Inflation in America stays at zero percent.
Unemployment in Europe goes from zero to 3 percent. And unemployment in America stays at zero percent.
Step three refers to the policy response. According to the Nash equilibrium there is a reduction in European money supply of 4 units and a reduction in American money supply of 2 units. Step four refers to the outside lag. Inflation in Europe goes from 3 to zero percent. Inflation in America stays at zero percent. Unemployment in Europe goes from 3 to 6 percent. And unemployment in America stays at zero percent. Table 3.2 gives an overview.
First consider the effects on Europe. As a result, given a supply shock in Europe, monetary interaction produces zero inflation in Europe. However, as a side effect, it raises unemployment there. Second consider the effects on America. As a result, monetary interaction produces zero inflation and zero unemployment in America. The initial loss of each central bank is zero. The supply shock in Europe causes a loss to the European central bank of 9 units and a loss to the American central bank of zero units. Then monetary interaction reduces the loss of the European central bank from 9 to zero units. And what is more, it keeps the loss of the American central bank at zero units.
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